When you behead a word, you remove the first letter and still have a valid word. You will be given clues for the two words, longer word first.
Example: Begin -> Sour, acidic
Answer: The words are Start and Tart.

When you curtail a word, you remove the last letter and still have a valid word. You will be given clues for the two words, longer word first.
Example: Begin -> Heavenly body
Answer: The words are Start and Star.